She had a briefcase with a bomb handcuffed to her wrist and had to chop off her arm to save herself.
Or a former cop out for revenge caught her, handcuffed her to a car with a leaky fuel tank and a jurry-rigged fuse then gave her a hack-saw.
Having cared for several cases, the loss of just one arm would be getting off easy.
If you’re trying to keep it believable, the injury should be a quick slice, not a crush. Large crush injuries cause secondary problems, like kidney failure and even multiple organ failure.
One has less blood loss with sudden, complete amputation. The blood vessels retreat into the tissue and constrict. There is blood loss, and first aid is necessay to live, but there is little or no secondary injury.
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She’s been thinking about it, and she’s sure she had it that morning, in the taxi, and she must have had it while shopping because she bought two bags of groceries…oh, well. She’s still hopeful that it will turn up.
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She lost it in a card game. A serious card game.
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She lost custody of the arm by rashly promising “I’d give my right arm for…” in a society that takes such promises literally, and enforces them surgically.
Some sort of ritualized combat with a member of an alien race, wherein the winner (that would be our heroine) has to have a limb amputated.
Don’t blame me if it’s weird. I’m trying to figure out how to attach things (i.e. wings) to a character.
One suggestion, cable snapping. It’s still far from unknown today, and happens with rigging, as well as tying ships to piers. Common, and can leave the character with a nice bit of mental whiplash:
If that so-and-so hadn’t screwed up with the hawser I’d still have two arms.
If I’d been half a foot closer it would have been my chest cut in half, not my arm.
I had nothing to do with the evolution, I was just walking by.
Am I lucky or unlucky?
•She tried to use a Craftsman Damaged Bolt/Nut Remover on an explosive bolt. (There were some unfortunate lapses in her training. Mistakes were made.)
•Was on a UN peacekeeping patrol in post-atomic Kashmir. Intel said the area was free of guerillas, which it was. They just forgot about the Tigers. (And since the fallout had killed off most of the local herbivores, they were very hungry Tigers.)
•Botched laser tattoo-removal.
•Partilal heat-shield failure on personal ballistic reentry capsule.
•Voluntarily had arm removed for cybernetic augmentation.
•She had an out of control amputation fetish. Which, tragically, peaked just months before Pfizer came out with psychiatric medication that would’ve cured it.
•Paradrop accident. (Maybe something with the static line wrapping around something it shouldn’t have.)
•Had one too many drinks, blacked out, woke up on a park bench two days later wearing a full Frank N. Further costume, wallet full of Yen notes, and arm burned off at the shoulder.
•Purposley blew off own arm to avoid deployment to the hopeless military quagmire in North Berkeley.
And…how futuristic are we talking here, BTW? Like, “mid 21st century” or “galaxy-spanning spacefaring civilization”?
Cosmetic amputation was fashionable for a brief period toward the end of the twenty-third centrury.
-Body-armour failure (micro-fissures) on the reinforced shoulder-cap.
-Sucked into the intake of a jet-engine.
-Struck by flying debris/micrometeor while performing EVA from a military space-shuttle doing repair work on space-based weapons systems.
-In a laser-sword duel with her cyborg father. Only, she didn’t know it was her father until after he cut off her arm and he made that admission to her. (Okay… I couldn’t resist)
-Force of an explosion (implosion?) threw her into the path of a plasma stream.
I need more detail. Are we going to leave a stump or take off the entire arm? Which part of the arm are we planning on leaving, if any? Lower arm with only the hand missing, or above the elbow type injury?
Freak Masturbation Accident.
It’s a lost arm were looking for, Mynn, not blindness! 
I question how well you know the character if you have to ask us for suggestions.
You say the subsequent events are important to her development but the injury itself isn’t. Every injury I received in the military was significant and how it happened shaped my development just as much (if not more) as subsequent events.
I suggest you visit with your character more and learn how it happened.
Injury sustained by minor antimatter leak while test piloting a new drive system.
First ever recorded quantum vacuum fluctuation to be apparent on the macroscopic scale; a section of her arm vanished and simultaneously reappeared a short distance away.
The arm was defleshed by a swarm of Threeks during landfall on Rigel 4 (don’t explain what Threeks are).
No fair! That was my idea!
Got really drunk at the office halloween party, her Walrus costume was the hit of the night, the funny thing was, no one could understand a word she was saying, which made it all the funnier. They kept the party going at some dive on Tatooine when some old fart with a lightsabre went and freaked out as she was trying to tell her drunk boyfriend to just back off and let the old fossil buy them all drinks. Sadly, no one understood a word she was saying.
HEY! Earthworm Jim!
Any o this sound good?
She got carried away with her new meat slicer. Then ate a sandwich.
She dropped her hover keys into a vat of liquid nitrogen and tried to retrieve them.
Anybody can go blind masturbating.
She lives about two miles inland on the Gulf Coast of Florida. One night, she reached into the back yard to bring her kitty into the house when an alligator…
It’s not wierd…there’d be some nice irony though if the alien race either had 28 arms…or they grew back in under a fortnight.